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Is it possible to measure the difference between an arduino and a bluetooth device to create a sort of wireless bluetooth leash? Let's say you have an arduino powered robot with bluetooth shield and a bluetooth enabled phone, could you have the robot move towards the phone if the distance separating them becomes greater than say 2 metres (no obstacles preventing movement)?

If you have a good measure of the RSSI vs distance of the wireless/bluetooth signal you could use that as a controlling factor. This would get you distance to the device, and a sort of home in on stronger signal to get back. If you use two antenna's you could triangulate the location of the transmitter and make better movements.
I would still keep a heartbeat signal just in case the signal strength was off as these are typically low power devices that sometime auto-adjust signal strength based on communication speed. That is to preserve power use only the gain needed to get desired data transmission rate. This could throw off the straight RSSI-distance method, but not the triangulation method.

A similar product that may give you a better idea of what's out there is the Stick N find.

It's a Bluetooth device that can detect the distance between your phone and the bluetooth sticker, although not very accurately, and there is about a 2 second delay. It cant do directional detection yet either. I suspect that your idea is a different spin on this same technology.

I would say wait on it a few years for the tech to develop and pull that one back out, see what's out there. Cool idea though!